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An Inclusion Theorem for Bohr-Hardy Summability Factors

Published online by Cambridge University Press:  20 November 2018

B. Thorpe
B. Thorpe*
Affiliation:
University of Birmingham, Birmingham, England University of Western Ontario, London, Ontario
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1. Let A denote a sequence to sequence transformation given by the normal matrix A = (ank )(n, k = 0, 1, 2, …), i.e., a lower triangular matrix with ann ≠ 0 for all n. For B = (bnk ) we write B ⇒ A if every B limitable sequence is A limitable to the same limit, and say that B is equivalent to A if B ⇒ A and A ⇒ B. If B is normal, then it is well known that the inverse of B exists (we denote it by B -l) and that B ⇒ A if and only if F = AB -1 is a regular transformation, i.e., transforms every convergent sequence into a sequence converging to the same limit. We say that a series ∑ an † is summable A if its sequence of partial sums is A-limitable.

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 23 , Issue 4 , August 1971 , pp. 653 - 658
Copyright
Copyright © Canadian Mathematical Society 1971

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